Yeah that heat exchanger and or the burners are plugged. DO NOT run it. Plugged = soot = carbon monoxide. With it being your rental I wouldn't dick with it. I would leave it to the land lords. Document document document. They(landlords) should've provided you with co detector per Megan's law, if you don't have one buy at least one with a digital read out. That exchanger and burners should be cleaned professionally. I recommend duct works, monster vac etc. I'd bet $5 after they clean the heat exchanger there's a crack. Cracked heat exchanger means new furnace. Pm me if you want my contact info for advice.